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In this lesson, we learn how to interpret limits whose values are unbounded, or infinite.

We’ll look at a few examples to show you how to compute these infinite limits.

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What is an Infinite Limit?

In calculus, a limit is the y-value to which a function approaches as the x-values get closer to some specified number. But what happens where the function doesn’t approach any particular y-value? In that case, we often say that the limit does not exist. However, in some situations, a more descriptive answer can be given. We say that f approaches ∞ as x approaches a number c if the values of f(x) become larger and larger without bound as x approaches c.

The notation for this situation is:

The limit as x approaches c is infinity.
Limit as x approaches c is infinity


On the other hand, we say that f approaches -∞ as x approaches c if the values of f(x) become unbounded in the negative sense (larger and larger negative values) as x approaches c. The notation is similar.

The limit as x approaches c is negative infinity.
In either situation, the function is said to have an infinite limit at the number x = c.The presence of an infinite limit in a function actually indicates that the function ‘blows up.’ You may have heard that nothing can travel faster than the speed of light.

Well, this is true in part because the function that measures the mass and energy of an object with respect to its velocity v has an infinite limit at the number v = c, where c in this case is the speed of light! The infinite limit tells us that mass/energy actually becomes unbounded as velocity approaches c; hence, no physical object can reach that speed.Now, let’s look at some examples.

A Case Study: 1/x

Consider the function f(x) = 1/x. What happens to the y-values when x approaches 0? The graph shows a vertical asymptote at x = 0, which is a good indication that we’re dealing with an infinite limit.

The function seems to shoot off the top of the graph when x approaches 0 from the right; that means the limit of f is ∞ as x → 0 from the right. The graph drops off the bottom of the graph when x approaches 0 from the left, so the limit of f is -∞ as x → 0 from the left.Let’s build a table of values to illustrate this behavior more concretely.

x (right of 0) f(x) = 1/x x (left of 0) f(x) = 1/x
1 1 -1 -1
1/2 2 -1/2 -2
1/4 4 -1/4 -4
1/8 8 -1/8 -8
0.01 100 -0.

01

-100
0.000 001 1,000,000 -0.000 001 -1,000,000

Notice how the y-values shown in the second column continue to increase without bound (100; 1,000,000; etc.), and the y-values shown in the fourth column decrease without bound (-100; -1,000,000; etc.). Some of the points in the table are plotted on the graph in the illustration.

Trace with your finger on the graph starting at the point (1,1). As you trace to the left, always staying on the graph, you eventually hit the top. The graph actually goes upward beyond what’s shown, and never stops! This is the essence of an infinite limit.

Graph of 1/x
Graph of 1/x

According to the table (or the graph), we conclude that:

Limit of 1/x

Evaluating Infinite Limits

We do have a quick way to work out infinite limits, but it only works for functions that look like fractions. That is, a limit as xc (from the right or from the left) of p(x)/q(x). (For other situations, you may want to use a graph or a table of values.)

  1. First try plugging in x = c.

    If the result is a nonzero divided by zero, then you know that the limit must be infinite (or sometimes, it does not exist at all). If the limit is not of the form nonzero/zero, then stop; this method does not apply.

  2. If the limit is one in which xc from the left, then pick a number a little smaller than c. If the limit is one in which xc from the right, then pick a number a little larger than c.
  3. Substitute your number for x. If the result is positive, then the limiting value is probably positive infinity (∞).

    If the result is negative, then the limiting value is probably negative infinity (-∞). In order to be 100% sure, plug in numbers even closer to c and make sure the results do indeed get unbounded and continue to have the proper sign.

Examples

Example 1

Find the limits:

Limit Example 1
Limit Example 1


(a) First plug in x = 2. The value is: (1 – 2)/(3(2) – 6) = -1/0, a nonzero/zero limit, which tells us that the limit is infinite. Since x is approaching 2 from the right (recall, that’s what the little + symbol means on the 2), pick a value greater than, but very close to 2, such as x = 2.01.

Plugging into the function, we get (1 – 2.01)/(3(2.01) – 6) = -33.

7 ; 0, so the limit is -;.(b) We already know this is an infinite limit from part (a), however this time x is approaching 2 from the left. Pick x = 1.99. We get (1 – 1.

99)/(3(1.99) – 6) = 33 ; 0, so the limit is ;.(In both cases, you should really check your work by plugging in numbers even closer to 2 from the right or from the left. I’ll leave this step to you.)

Example 2

Find the limit:

Limit Example 2
This example doesn’t look like a fraction, but actually it is! 6 csc x = 6 / sin x. Now since sin 0 = 0, we know we’re in an infinite limit (nonzero/zero). Approaching from the right, sin x is positive, so 6 / sin x is positive, and this shows that the limit is positive infinity, or ∞.

Lesson Summary

As xc, if the values of a function get larger and larger without bound in the positive or negative sense, then the function has an infinite limit at c. Use a graph or table of numbers to decide if the function has an infinite limit and whether it is ∞ or -∞.

If the limit has the form nonzero/zero, then determine the sign of the limit (∞ or -∞) by plugging in numbers very close to c.

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